Optimization Problem - Calculus

In summary, the problem is to find the dimensions of a cylindrical tin can with a volume of 1000cm3 that requires the minimum amount of tin. The solution involves isolating a variable, which leads to a relationship between r and h. The surface area function is then rewritten as a function of one variable, and the normal technique for finding the minimum or maximum value is applied.
  • #1
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Homework Statement



A cylindrical shaped tin can must have a volume of 1000cm3.

Find the dimensions that require the minimum amount of tin for the can (Assume no waste material). The smallest can has a diameter of 6cm and a height of 4 cm.

Homework Equations



[tex]V = \pi r^{2}h[/tex]

[tex]P = 2( \pi r^{2}) + (2 \pi r)h[/tex]

The Attempt at a Solution



I basically start off by isolating a variable:

[tex]V = \pi r^{2}h[/tex]

[tex]1000 = \pi r^{2}h[/tex]

[tex]318.3 = r^{2}h[/tex]

[tex]r^{2} = \frac{318.3}{h}[/tex]
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This is where I'm stuck. Do I plug this into the surface area formula.
I know I have to sub this into something else, and then expand and find the derivative of that new equation to find where h > 4.
Just need to find out what to do next from here.

Thanks for any help!
 
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  • #2
You have a relationship between r and h (but leave pi in - 1000/pi is not exactly equal to 318.8). Now rewrite your surface area function P (why is it P?) so that it is a function of one variable, either r or h. Then use the normal technique for finding the minimum or maximum function value.
 

1. What is an optimization problem in calculus?

An optimization problem in calculus involves finding the maximum or minimum value of a function, subject to certain constraints. This can be done by using techniques such as differentiation and setting up equations to solve for the optimal value.

2. How do you approach an optimization problem in calculus?

To solve an optimization problem in calculus, you first need to identify the objective function and the constraints. Then, you can use techniques such as differentiation and setting up equations to find the optimal value that satisfies the constraints.

3. What is the significance of optimization problems in real life?

Optimization problems in calculus have many real-life applications, such as determining the most efficient way to use resources, maximizing profits, and minimizing costs. They are also commonly used in engineering, economics, and other fields.

4. What are some common techniques used to solve optimization problems in calculus?

Some common techniques used to solve optimization problems in calculus include the use of derivatives, setting up equations, and using the first and second derivative tests to determine if a critical point is a maximum or minimum value.

5. Can you provide an example of an optimization problem in calculus?

One example of an optimization problem in calculus is finding the dimensions of a rectangular box with a fixed volume and minimum surface area. This involves setting up an equation for the surface area and using differentiation to find the minimum value.

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